Individual students learn in different ways. When manipulatives are used, the senses are brought into learning: students can touch and move objects to make visual representations of mathematical concepts. Manipulatives can be used to represent both numbers and operations on those numbers. In addition to meeting the needs of students who learn best in this way, manipulatives afford the teacher new ways of visiting a topic.
A sound lesson on any mathematical topic should involve multiple instructional methods. Incorporating several different instructional techniques increases the possibility that all students will develop mathematical understanding through at least one method. One model, discussed by Suzanne Alejandre in her position paper "Middle School Mathematics Curriculum," describes combining the use of manipulatives, technology and paper/pencil activities.
"Through presenting an activity with the three components (activity, technology, formalizing) we not only give students with different learning styles different ways to see a problem, we give them the extra time they may require for learning. When I think in terms of varied presentations of the same problem, I can't help but allot more time to the material. Time and experience in class enrich an activity: students can learn from their experiences and connect the mathematics to those experiences. Such a foundation helps them to understand and appreciate mathematics."
In her article "Math Curse or Math Anxiety?" (hosted by the National Council of Teachers of Mathematics [NCTM]), Vanessa B. Stuart provides data about the usefulness of manipulatives as a tool for instruction.
"As manipulatives and cooperative groups become more widely used in mathematics classes, I wanted to know whether students perceived these aids and situations as being useful learning tools. Three-fourths of the students thought that using manipulatives when learning a new mathematical concept was helpful. Most of the comments indicated that using manipulatives first helped students see the origin of the numbers in the formulas. Fewer than one-fourth of the students said that manipulatives were not helpful learning tools, stating that they were confusing."
Near the end of her article, Stuart states:
"Williams (1988, 101) paraphrased a Chinese proverb: 'Tell me mathematics and I forget; show me mathematics and I may remember; involve me... and I will understand mathematics. If I understand mathematics, I will be less likely to have math anxiety. And if I become a teacher of mathematics, I can thus begin a cycle that will produce less math anxious students for the generations to come.'"
Although the use of manipulatives in combination with other methods can enrich and deepen students' understanding, relying only on manipulatives as a means of instruction can also be ineffective. Students may lose the opportunity for deeper conceptual learning if manipulatives are used without further formal discussion, abstraction, and mathematical connection.
Resources on the Web:
Math Forum: Internet Mathematics Library - Manipulatives
This comprehensive catalog provides a page of links to math education sites that discuss the use of manipulatives in the classroom, and many pages of annotated math sites to browse and search for manipulatives. ABOUT A MILLION WEB RESOURCES http://mathforum.org/library/
The Lesson Plans Page by Kyle Yamnitz, Univ. of Missouri
A searchable archive of lesson plans for arithmetic, patterns, basic operations (addition, subtraction, multiplication, division), time, estimation, graphs, place value, fractions, measurement, number sense, and many more. Includes suggestions for ...more>> http://www.lessonplanspage.com/math-htm/?grade=grades-2-3
Hand Made Manipulative Instructions by Margo Lynn Mankus
Pattern block cutouts: triangles, squares, smaller and larger rhombuses, trapezoids, and hexagons. Base-ten blocks in sets of ones, tens, and hundreds; and base-five blocks in sets of ones, fives, and twenty-fives. XY-block sets of X, X2, XY, Y, and ...more>> http://mason.gmu.edu/~mmankus/Handson/manipulatives.htm
Math Education Page: Manipulatives by Henri Picciotto
The site includes discussion, debate, sample activities and lessons on the use of manipulatives. ...more>> http://www.mathedpage.org/index.html